Eastern Illinois University—Department of Early Childhood, Elementary and Middle Level Education
The Department Lesson Plan
Elements of the Department Lesson Plan are meant to be adapted for the following strategies: Direct Instruction, Concept Teaching, Cooperative Learning, Problem-Based Instruction, Classroom Discussion, and Guided Inquiry.
TITLE OF THE PLAN: Collecting Data GRADE LEVEL: 8th Grade Algebra SUBJECT AREA: Mathematics CONCEPT/SKILL: Collecting Data TARGET AUDIENCE: Regular class TIME FRAME: One Class Period (50 Minutes)
I. PREPARING TO TEACH: Identifying goals, objectives, purpose, and gathering materials and resources.
A. GOALS:
Use random sampling to draw inferences about a population.
oCCSS.MATH.CONTENT.7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
oCCSS.MATH.CONTENT.7.SP.A.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Draw informal comparative inferences about two populations.
oCCSS.MATH.CONTENT.7.SP.B.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
Investigate chance processes and develop, use, and evaluate probability models.
oCCSS.MATH.CONTENT.7.SP.C.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
oCCSS.MATH.CONTENT.7.SP.C.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
oCCSS.MATH.CONTENT.7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
oCCSS.MATH.CONTENT.7.SP.C.7.A Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
oCCSS.MATH.CONTENT.7.SP.C.7.B Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
oCCSS.MATH.CONTENT.7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
oCCSS.MATH.CONTENT.7.SP.C.8.A Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
oCCSS.MATH.CONTENT.7.SP.C.8.B Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
oCCSS.MATH.CONTENT.7.SP.C.8.C Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
B. OBJECTIVE(S):
Students will be able to construct and read a frequency table and then make inferences about the frequency of events based on the data provided in the table.
Students will be able to identify all of the properties of a frequency table i.e. sampling, data, population, sample, tally marks, and cumulative frequency.
Students will be able to identify good and bad sampling in a statistical situation.
Students will be able to start to explore advanced algebraic equations through the exploration of frequency tables and the data that is provided in them.
C. PURPOSE: The purpose of this lesson is to teach students how to read frequency tables. Students will also be able to take a set of data and create frequency tables. Upon creating the frequency tables or reading the frequency tables that have been given to them, students will be able to analyze the data and create equations to answer questions about the data.
D. MATERIALS: Teacher’s Manual, Projector/Computer, Power Point, Balloons
E. RESOURCES:
Cummins, J. (2004). Collecting Data. In Glencoe Algebra: Concepts and applications (pp.32-37). New York, New York: Glencoe/McGraw Hill.
II. INSTRUCTIONAL PROCEDURES: This is the heart of the lesson. Include detailed step-by-step bulleted or numbered procedures. The procedures include: information (concepts/content/skills), activity directions, leading question(s), examples, questions and expected answers, etc.
A. FOCUSING ACTIVITY:
a. Quick Questions
i. Travis is 2 years older than his sister Mickie. Together their ages total 106. How old is each person? Travis 54, Mickie 52 ii. Mack has $300 in the bank at an annual interest rate of 3%. How much money will Mack have in the bank in four years? $336 iii. How many ways are there to make $0.75 using dimes and nickels? 8 Ways iv. The table shows the cost of living in an apartment for one year.
Expense
Apt. A Cost ($)
Apt. B Cost ($)
Monthly Rent
495
515
Average Electricity Per Month
65
23
Average Gas Per Month
0
15
1. Which apartment has the lower monthly rent? Apartment A
2. Overall, which apartment is cheaper to live in? Apartment B
B. PURPOSE: Today, we are going to be learning about collecting data and how we organize that data so that we can read and analyze it. This can be useful in many different situations outside of the classroom especially when you are conducting surveys such as “What is your favorite pizza topping” or just seeing if there is a general interest for something like “Should schools have soda machines?”
§These problems will be written on the board or projected onto a smartboard screen for the kids to work on in the first couple of minutes of class
oGo ahead and find your seat and get to working on the quick questions so that we can discuss them in a few moments.
oAfter 5 minutes, I will conduct a discussion of each of the answers for the questions. These questions are designed to get the kids thinking about mathematics and start them in reading data before we start the lesson on it.
Define word concepts
oSampling: a convenient way to gather data
oData: information
oPopulation: a group of things
oSample: a small group that is used to represent a much larger population
Define what a good sample is
oRepresentative of the larger population
oSelected at random
oLarge enough to provide accurate data
§Surveys can be biased and give false data if the criteria above are not followed. There is no criteria on what “large enough” is for accurate data, so you have to look at each situation individually to decide if there is good data or not.
Example 1: One hundred people in Lafayette, Colorado, were asked to eat a bowl of oatmeal every day for a month to see whether eating a healthy breakfast daily could help reduce cholesterol. After 30 days, 98 of those in the sample had lower cholesterol. Is this a good sample? Explain.
oIf the people were randomly chosen, then this is a good sample. Also, the sample appears to be large enough to be representative of the population. For example, the results of two or three people would not have been enough to make any conclusions.
Your Turn: Determine whether each is a good sample. Explain.
oTwo hundred students at a school basketball game are surveyed to find the students’ favorite sport. No; many of those surveyed would prefer basketball.
oEvery other person leaving a supermarket is asked to name their favorite soap. Yes; it is random
Definitions
oFrequency Table: a table that organizes data
oTally Marks: indicates the frequency of events
§Go through how tallies are done and what they represent.
Example 2: In an experiment, students “charged” balloons by rubbing them with wool. Then the students placed the balloons on a wall and counted the number of seconds they remained. The class results are shown in the chart at the right. Make a frequency table to organize the data.
oStep 1: Make a table with three columns: time (s), tally, and frequency. Add a title
oStep 2: It is sometimes helpful to use intervals so there are fewer categories. In this case, we are using intervals of size 10.
oStep 3: Use tally marks in each row and record this number in the Frequency column.
Static Electricity Time (s)
15
52
26
22
25
26
29
33
36
20
43
21
30
39
34
35
27
29
42
35
16
18
21
21
40
Static Electricity
Time (s)
Tally
Frequency
15-24
8
8
25-34
9
9
35-44
7
7
45-54
1
1
Your Turn
oMake a frequency table to organize the data in the chart.
Noon Temperature
32
30
18
29
20
14
21
32
36
15
19
10
16
22
25
30
26
21
Noon Temperature
Temperature (°C)
Tally
Frequency
5-14
2
2
15-24
8
8
25-34
7
7
35-44
1
1
Definitions
oCumulative Frequency Table: frequencies are accumulated for each item
§Suppose the science teacher wanted to know how many balloons stayed on the wall no more than 44 seconds.
Static Electricity
Time (s)
Frequency
Cumulative Frequency
15-24
8
8
25-34
9
17
35-44
7
24
45-54
1
25
**From the cumulative frequency table, we see that 24 balloons stayed on the wall for 44 seconds or less. Or, 24 balloons stayed on the wall for no more than 44 seconds.
Check – Up!
oExplain the difference between a frequency table and a cumulative frequency table.
§A frequency table shows the number of times a single event occurs. A cumulative frequency table shows the number of times an event occurs plus the previous events.
oList some examples of how a survey might be biased.
§A sample might be biased if there are too few items in the sample or if the sample is not random.
Example 3: (Analyzing the information after it has been put into a chart). Problem: Owners of a restaurant are looking for a new location. They counted the number of people who passed by the proposed location one afternoon. The frequency table shows the results of their sampling.
oWhich two groups of people passed by the location most frequently? Adults in their 30s and 40s
oIf the restaurant is an ice cream shop aimed at teens during their lunchtimes, is this a good location for the restaurant? Explain. Since very few teens pass by the location compared to adults, the owners should probably look for another location.
Age of People
Tally
Frequency
Under 13
7
7
Teens
10
10
20s
18
18
40s
36
36
50s
19
19
60s
11
11
Check – Up!
oDetermine whether each is a good sample. Explain.
§Four people out of 500 are randomly chosen at a senior assembly and surveyed to find the percent of seniors who drive to school. No, the sample is not large enough
§Six hundred randomly chosen pea seeds are used to determine whether wrinkled seeds or round seeds are the more common type of seed. Yes
Quiz (Pull out a piece of paper and number it to correspond to the quiz on the board)
oDetermine whether each is a good sample. Describe what caused the bias in each poor sample. Explain.
§Thirty people standing in a movie line are asked to name their favorite actor. No; the people might be in line because the movie has a favorite star.
§Police stop every fifth car at a sobriety checkpoint. Yes; the sample is random and representative of drivers in the area
§Every other household in a neighborhood of 240 homes is surveyed to determine how many people in the area recycle. Yes; the sample is random and representative of people in the neighborhood.
oRefer to the chart.
§Make a frequency table to organize the data.
§What number of goals was scored most frequently? 2 goals
§How many times did the team score 8 goals? 1 time
§How many more times did the soccer team score six goals than three goals? 2 times
Number of Soccer Goals Scored This Season
1
2
5
1
6
2
6
8
4
2
4
5
5
1
3
4
7
2
2
6
4
Answer:
Number of Goals
Tally
Frequency
1
3
3
2
5
5
3
1
1
4
4
4
5
3
3
6
3
3
7
1
1
8
1
1
MODELING: Students will have the opportunity to interact with each other and their instructor as they go through this lesson. The lesson has been set up in a way that allows for the students to create charts, come to solutions using different strategies, as well as discover new ways of accomplishing the task of analyzing, collecting, and displaying data.
CHECKING FOR UNDERSTANDING: All of the sections labeled as “Check-Up” are sections where I will be checking the students for their understanding of the material.
D. TASK/GUIDED PRACTICE: Students will follow along as we go through the lesson plan. There are sections throughout this lesson that are designed for student input. The students will have a large role in the success of this lesson. The teacher will simply act as a guide to get the students through and understanding the material.
E. ACCOMMODATIONS/ADAPTATIONS: The accommodations and adaptations in this lesson will vary depending on the need of the student. I can create handouts with the graphs already gridded out on their page so all they have to do is put in the numbers, I can create a grid with just some of the numbers missing, and I can even make a page with all of the work done. This section, as stated before, will really depend on the need of the students in my class. Power points will be used and can be printed for further assistance.
F. INDEPENDENT PRACTICE (include when appropriate): Instead of using the data in the experiment with the balloon, a teacher might consider having the kids do their own experiment with the balloons and then exchange data. This would be fun, exciting, and educational for the kids so that they can see how the concepts that they are learning apply to the world outside of their classroom.
III. CLOSURE: We will go through all of the definitions and review tallies, frequency charts, cumulative frequency charts, and data reading skills.
IV. ASSESSMENT
A. Student Assessment: The Quick Questions at the beginning of the period will allow for me to see how much the kids understand how charts work and how to read them in general. Throughout the lesson, I will quiz the kids on what they know with the “Check-Ups”. At the end of the lesson, I will further test the knowledge of my students with a quiz.
The Department Lesson Plan
Elements of the Department Lesson Plan are meant to be adapted for the following strategies: Direct Instruction, Concept Teaching, Cooperative Learning, Problem-Based Instruction, Classroom Discussion, and Guided Inquiry.
TITLE OF THE PLAN: Collecting Data
GRADE LEVEL: 8th Grade Algebra
SUBJECT AREA: Mathematics
CONCEPT/SKILL: Collecting Data
TARGET AUDIENCE: Regular class
TIME FRAME: One Class Period (50 Minutes)
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
- E. RESOURCES:
Cummins, J. (2004). Collecting Data. In Glencoe Algebra: Concepts and applications (pp.32-37). New York, New York: Glencoe/McGraw Hill.- A. FOCUSING ACTIVITY:
- a. Quick Questions
i. Travis is 2 years older than his sister Mickie. Together their ages total 106. How old is each person? Travis 54, Mickie 52ii. Mack has $300 in the bank at an annual interest rate of 3%. How much money will Mack have in the bank in four years? $336
iii. How many ways are there to make $0.75 using dimes and nickels? 8 Ways
iv. The table shows the cost of living in an apartment for one year.
- Quick Questions (5-10 minutes)
- o (See information above)
- § These problems will be written on the board or projected onto a smartboard screen for the kids to work on in the first couple of minutes of class
- o Go ahead and find your seat and get to working on the quick questions so that we can discuss them in a few moments.
- o After 5 minutes, I will conduct a discussion of each of the answers for the questions. These questions are designed to get the kids thinking about mathematics and start them in reading data before we start the lesson on it.
- Define word concepts
- o Sampling: a convenient way to gather data
- o Data: information
- o Population: a group of things
- o Sample: a small group that is used to represent a much larger population
- Define what a good sample is
- o Representative of the larger population
- o Selected at random
- o Large enough to provide accurate data
- § Surveys can be biased and give false data if the criteria above are not followed. There is no criteria on what “large enough” is for accurate data, so you have to look at each situation individually to decide if there is good data or not.
- Example 1: One hundred people in Lafayette, Colorado, were asked to eat a bowl of oatmeal every day for a month to see whether eating a healthy breakfast daily could help reduce cholesterol. After 30 days, 98 of those in the sample had lower cholesterol. Is this a good sample? Explain.
- o If the people were randomly chosen, then this is a good sample. Also, the sample appears to be large enough to be representative of the population. For example, the results of two or three people would not have been enough to make any conclusions.
- Your Turn: Determine whether each is a good sample. Explain.
- o Two hundred students at a school basketball game are surveyed to find the students’ favorite sport. No; many of those surveyed would prefer basketball.
- o Every other person leaving a supermarket is asked to name their favorite soap. Yes; it is random
- Definitions
- o Frequency Table: a table that organizes data
- o Tally Marks: indicates the frequency of events
- § Go through how tallies are done and what they represent.
- Example 2: In an experiment, students “charged” balloons by rubbing them with wool. Then the students placed the balloons on a wall and counted the number of seconds they remained. The class results are shown in the chart at the right. Make a frequency table to organize the data.
- o Step 1: Make a table with three columns: time (s), tally, and frequency. Add a title
- o Step 2: It is sometimes helpful to use intervals so there are fewer categories. In this case, we are using intervals of size 10.
- o Step 3: Use tally marks in each row and record this number in the Frequency column.
Static Electricity Time (s)Static Electricity
- Your Turn
- o Make a frequency table to organize the data in the chart.
Noon TemperatureNoon Temperature
- Definitions
- o Cumulative Frequency Table: frequencies are accumulated for each item
- § Suppose the science teacher wanted to know how many balloons stayed on the wall no more than 44 seconds.
Static Electricity- Check – Up!
- o Determine whether each is a good sample. Explain.
- § Four people out of 500 are randomly chosen at a senior assembly and surveyed to find the percent of seniors who drive to school. No, the sample is not large enough
- § Six hundred randomly chosen pea seeds are used to determine whether wrinkled seeds or round seeds are the more common type of seed. Yes
- Quiz (Pull out a piece of paper and number it to correspond to the quiz on the board)
- o Determine whether each is a good sample. Describe what caused the bias in each poor sample. Explain.
- § Thirty people standing in a movie line are asked to name their favorite actor. No; the people might be in line because the movie has a favorite star.
- § Police stop every fifth car at a sobriety checkpoint. Yes; the sample is random and representative of drivers in the area
- § Every other household in a neighborhood of 240 homes is surveyed to determine how many people in the area recycle. Yes; the sample is random and representative of people in the neighborhood.
- o Refer to the chart.
- § Make a frequency table to organize the data.
- § What number of goals was scored most frequently? 2 goals
- § How many times did the team score 8 goals? 1 time
- § How many more times did the soccer team score six goals than three goals? 2 times
Number of Soccer Goals Scored This SeasonAnswer:
MODELING: Students will have the opportunity to interact with each other and their instructor as they go through this lesson. The lesson has been set up in a way that allows for the students to create charts, come to solutions using different strategies, as well as discover new ways of accomplishing the task of analyzing, collecting, and displaying data.
CHECKING FOR UNDERSTANDING: All of the sections labeled as “Check-Up” are sections where I will be checking the students for their understanding of the material.